Complex analysis and geometry

Purdue University, West Lafayette IN

Investigators

Abstract

One role of mathematics is to provide the terms in which to describe the world around us. As we are discovering and dealing with more and more complicated phenomena, both natural and societal, it is critical that the description nevertheless stay simple. Mathematics achieves this by introducing new notions. Here is an example, pertinent for this project. Quantities in the real world are measured by real numbers, and since the invention of analytic geometry we know that real world figures---curves, surfaces, etc.---can be described by functions of real variables. Yet real world quantities in oscillatory phenomena, for example voltage in an alternating current, can be described much more simply in terms of complex numbers. It takes some investment to introduce complex numbers, but once done, the description becomes fully transparent. By now we understand well that complex numbers are indispensable in a vast number of problems that arise in science and engineering. Similarly, it is often advantageous to pass from functions of real variables and the figures they describe to functions of complex variables and the associated geometric figures. This research will deal with fundamental properties of functions of complex variables and of complex geometric figures, known as complex manifolds. One component is motivated by problems that arise in the quantum description of the micro-world, and seeks to understand to what extent this description is independent of the somewhat arbitrary choices one is forced to make as the mathematical description is constructed. It turns out that this problem and various others that arise in the study of manifolds have a common generalization, and the answer promises to depend on the notion of curvature. The PI will study curvature in various situations; it is the concept that unifies the three components of this research.---As said, some of the concrete problems in the project are directly motivated by quantum theory, and have a potential to impact theoretical physics. By involving graduate students, the project will also serve to introduce young people to mathematical research. In more technical terms, one component of this project studies fields of Hilbert spaces, generalizations of hermitian vector bundles. They arise as direct images of holomorphic vector bundles, and the PI will seek to connect the curvature of the direct image with the curvature of the vector bundle. A second component is motivated by the problem of Kahler metrics of constant scalar curvature. The existence and uniqueness of these metrics is related to the geometry of the space of all Kahler metrics, an infinite dimensional Riemannian manifold, in particular to the geodesics in this space. The PI will study these geodesics and the partial differential equation that governs them. The third component is about general hermitian metrics in holomorphic vector bundles and their curvature. The metric can be singular and the bundle of infinite rank. The questions here concern how to generalize to this setting the phenomena well understood in the case of line bundles.